farewel song
TRANSCRIPT
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1
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
CHEMISTRY PAPER-I
PART I CHEMISTRY
SECTION I (TOTAL MARKS: 21)
(SINGLE CORRECT CHOICE TYPE)
This section contains 7 multiple choice questions. Each question has four choices (a),(b), (c) and (d) outof
which ONLY ONE is correct.
1. Dissolving 120g of urea (mol. wt 60) in 1000g of water gave a solution of density 1.15g/mL. The
molarity of the solution is
(a) 1.78 M (b) 2.00 M (c) 2.05 M (d) 2.22 M]
1. (c)
2. AgNO3
(aq.) was added to an aqueous KCl solution gradually and the conductivity of the solution
was measured. The plot of conductance versus the volume of AgNO3 is
(a) (P) (b) (Q) (c) (R) (d) (S)
2. (d) AgNO KCl AgCl KNO 3
3
Conductivity will remain same for some time then gradually increase due to excess of
KCl.
3. Among the following compounds, the most acidic is
(a) p-nitrophenol (b) p-hydroxybenzoic zcid
(c) ohydroxybenzoic acid (d) p-toluic acid
3. (c) Due to ortho effect 2
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respectively are
(a) Octahedral, tetrahedral and square planar
(b) Tetrahedral, square planar and octahedral
(c) Square planar, tetrahedral and octahedral
(d) Octahedral, square planar and octahedral
6. (b)
2
3
NiCl sp
4
2
2
Ni CN dsp
4
2
3 2
2 6
Ni H O sp d
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3
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
7. Bombardment of aluminium by
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(b)
3 2
An NO (c)
3 3
Al NO (d)
3 2
Pb NO
15. (a)
16. The final solution contains
(a)
2 2
3 4 4
Pb NH and CoCl
(b)
3 2
3 3 4 4
Al NH and Cu NH
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(c)
2
3 3 2
Ag NH and Cu NH 4
(d)
2
3 3 2 6
Ag NH and Ni NH
16. (c)7
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H-108, New Asiatic Building, Connaught place, New Delhi 110001
SECTION IV (TOTAL MARKS: 28)
(INTEGER ANSWER TYPE)
This section contains 7 questions. The answer to each of the questions is a single-digit integer, ranging
from 0 to 9. The bubble corresponding to the correct answer is to be darkened in the ORS.
17. The work functionof some metals is listed below. The number of metals which will show
photoelectric effect when light of 300nm was wavelength falls on the metals is
Metal Li Na K Mg Cu Ag Fe Pt W
eV 2.4 2.3 2.2 3.7 4.8 4.3 4.7 6.3 4.75
17. (4)
o
3000A 4.1 eV.
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All below this value will eject photo electron
18. To an evacuated vessel with movable piston under external pressure of 1atm, 0.1mol of He and 1.0
mol of an unknown compound (vapour pressure 0.68 atm. At 0
o
C) are introduced. Considering the
ideal gas behaviour, the total volume (in lite) of the gases at 0
o
C is close to.
18. (7)
1 273
0.1 7
12 0.32
19. Reaction of Br2 with Na2CO3 in aqueous solution gives sodium bromide and sodium bromated
with evolution of CO2
gas. The number o sodium bromide molecules involved in the balanced
chemical equation is
19. (5)
2 2 3 3 2
3Br 3Na CO 5NaBr NaBrO 3CO
20. The difference in the oxidation numbers of the two types of sulphur atoms in Na S O is
2 4 6
20. (5)
5 0 = 5
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rotated on a horizontal circular path about vertical axis. The maximum tension that the string can
bear is 324 N. The maximum possible value of angular velocity of ball (in radian/s) is:
(a) 9 (b) 18 (c) 27 (d) 36
25. (d) T
2 2
sin m r m Lsin
2
T m L
T 324
18 / 0.5 36
mL 0.5 0.5
10
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
26. A meter bridge is set up as shown, to determine an unknown resistance X using a standard 10
ohm resistor. The galvanometer shows null point when tapping key is at 52 cm. mark. The end
corrections are 1 cm and 2 cm respectively for the ends A and B. The determined value of X is
(a) 10.2 ohm (b) 10.6 ohm (c) 10.8 ohm (d) 11.1 ohm
26. (b) x /10 52 1 / 48 2 x 10.6
27. A 2 -F capacitor is charged as shown in figure. The percentage of its stored energy dissipated
after the switch S is turned to position 2 is
(a) 0% (b) 20% (c) 75% (d) 80%
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% of its stored energy dissipated =
2
2
Q 4
100 80%
5 Q
28. A police car with a siren of frequency 8 kHz is moving with uniform velocity 36 km/hr towards a
tall building which reflects the sound waves. The speed of sound in air is 320 ms
-1
. The frequency
of the siren heard by the car driver is:
(a) 8.50 kHz (b) 8.25 kHz (c) 7.75 kHz (d) 7.50 kHz
28. (a) Velocity of sound, C = 320 ms
-1
frequency of the siren heared by the car driver
f =
0
f C V / C V 8 320 10 / 320 10 8.50kHz
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This section contains 4 multiple choice questions. Each question has four choices (a), (b), (c) and (d) out
of which ONE OR MORE may be correct.
31. A metal rod of length L and mass m is pivoted at one end. A thin disk of mass M and radius
R ( Angular frequency for case B
(d) Angular frequency for case A < Angular frequency for case B
31. (a, d) Torque due to weight is same in both case (A)
T I 1 / I
sys sys
Case A: When disc is not rotating then M.I of system about point of suspension = IA
Case B : When disc is rotating then M.I. of system about point of suspension = IB
2
pivot pivot Pivot
B rod disc A rod disc
MR
I I I I I I
2
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inside
E 0 from Gauss law A A
2
A B A A B B
A B 2 2
A B B B A A
Q Q Q R R
, C
4 R 4 R Q R R
$
$ $
# # $
on surface on surface
E & E A A B B $ $
on surface on surface E < E D A B14
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
SECTION-III (TOTAL MARKS: 15)
(PARAGRAPH TYPE)
This section contains 2 paragraphs. Based upon the first paragraph 2 multiple choice questions andbased
on the paragraph 3 multiple choice questions have to be answered. Each of these questions has four
choices (a), (b), (c) and (d) out of which ONLY ONE is correct.
Paragraph for questions 35 to 36
A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids
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containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let N bethe
number density of free electrons, each of mass m. When the elec trons are subjected to an electricfield,
they are displace relatively away from the heavy positive ions. If the electric field becomes zero, the
electrons begin to oscillate about the positive ions with a natural angular frequency p
, which is called
the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied thathas
an angular frequency , where a part of the energy is absorbed and a part of it is reflected. As
appropriate p
, all the free electrons are set to resonance together and all the energy is reflected. This is the
explanation of high reflectivity of metals.
35. Taking the electronic charge as e and the permittivity as
0
% , use dimensional analysis to
determine the correct expression for p
.
(a)
0
Ne
m%
(b)
m 0
Ne
%
(c)
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2
1/2 1/2
p 0 Ne m
%
2
p
0
Ne
m
%15
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
36. Estimate the wavelength at which plasma reflection will occur for a metal having the density of
electrons
27 3
N 4 10 m
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, where these quantities are in proper SI
units.
(a) 800 nm (b) 600 nm (c) 300 nm (d) 200 nm
36. (b) f / 2 #
2
27 19
30 11
4 10 1.6 10
10 10
2
19 68 15 15
1.6 10 4 10 2 1.6 10 3.2 10
8
7
15
3 10 6
c / f 10 m
3.2 10 3.2
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1 2
E 4E 1 2
39. Consider the spring mass system, with the mass submerged in water, as shown in the figure. The
phase space diagram for one cycle of this system is:
39. (b) The mass has zero momentum at highest position. From here, it moves down, during
which position decreases but momentum increases in negative value . After one cycle,
its position is slightly less than initial value (damped oscillation). 17
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
SECTION IV (TOTAL MARKS: 28)
(INTEGER ANSWER TYPE)
This section contains 7 questions. The answer to each question is a single digit integer ranging from 0 to9.
The bubble corresponding to the correct answer is to be darkened in the ORS.
40. Steel wire of length L at 40
o
C is suspended from the ceiling and then a mass m us hung from
its free end. The wire is cooled down from 40
o
C to 30
o
C to regain its original length L. The
coefficient of linear thermal expansion of the steel is 10
-5
/
o
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6
m 10 kg 1 mg
42. A block is moving on an inclined plane making an angle 45
o
with the horizontal and the
coefficient of friction is - . The force required to just push it up the inclined plane is 3 times the
force required to just prevent it from sliding down. If we define N = 10 -, then N is
42. (5)
u
d u d
F mg sin cos
F mg sin cos & F 3F
1 / 2 tan 1 / 2
N 10 5
-
-
-
- 18
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
43. A boy is pushing a ring of mass 2 kg and radius 0.5m with a sticks as shown in the figure. The
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2
0 0
2
A / LR 300 I sin 300t
N 0.6 6
- -
#
46. Four point charges, each of +q, are rigidly fixed at the four corners of a square planar soap fil of
side a. The surface tension of the soap film is . The system of charges and planar film asre in
equilibrium, and
1/ N
2
q
a k
) *
, where k is a constant . then N is
46. (3) Consider the upper half of the setup
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For equilibrium of the system
2 2
2 2
0 0
1 q 1 q 1
2. a 2 . . .
4 4 a 2
2a
) *
#% #%
3 2
0
a q 1 / 4 . 1 1/ 2 2 #%
) *
a = constant
1/3
2
q / N 3
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" 5 . If
n n
n
a
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6 2 0
2
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2 6
and
2
5 5 6 2 0
2 5 5 2 6
10 10 8 8 8 2 8 2
8 8
10 8
9 9 9 9 9 9
9
a 2a 2 2 2 6 6
3
2a 2 2 2
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5
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5
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5 5
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5 5
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5
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521
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
50. A straight line L through the point (3, -2) is inclined at an angle 60
o
to the line 3x y 1 . If L
also intersects the x axis, then the equation of L is
(a) y 3x 2 3 3 0 (b) y - 3x 2 3 3 0
(c) 3y x 3 2 3 0 (d) 3y x 3 2 3 0
50. (b) If required line has slope = m
m 3 o
tan 60 m 0, 3
1 3m
Since, the line intersects the x axis
equations is, y 2 3 x 3 y 3x 2 3 3 0
51. Let
0 0
x , y be the solution of the following equations
ln 2 ln 3
2x 3y
ln x ln y
3 2
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Then x0 is
(a) 1/6 (b) 1/ 3 (c) (d) 6
51. (c) Since ln 2 ln 2 ln x ln 3 ln 3 ln y and lnx . ln 3 ln y. ln 2 = 0
2 2
ln 2 ln x ln y ln 3 ln 3 ln 2 .(i)
and ln x.ln 3 ln y.ln 2 0 (ii)
Using (i) ln 2 (ii) ln 3, we get
2 2 2 2
ln x. ln 2 ln 3 ln 2 ln 3 ln 2
ln x ln 2 x 1 / 2
52. The value of
ln 3 2
2 2
ln 2
x sin x
dx
sin x sin ln 6 x
4
is
(a) 1 / 4 ln 3 / 2 (b) 1 / 2 ln 3 / 2 (c) ln 3 / 2 (d) 1 / 6 ln 3 / 2
52. (a)
ln 3 2
2 2
ln 2
x sin x
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I dx
sin x sin ln 6 x
4
2
x t 2xdx=dt
ln 3
ln 2
1 sin t
I dt
2 sin t sin ln 6 t
4
Using
b b
a a
1
f x dx f x f a b x dx
2
4 4
ln 3
ln 2
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1 sin t sin ln 6 t
I dt
4 sin t sin ln 6 t sin ln 6 t sin t
4
[since ln 3 + ln 2 = ln 6]
ln 3
ln 3
ln 2
ln 2
1 1 1 3
I dt t ln
4 4 4 2
422
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
53. Let
a i j k, b = i - j + k
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and
c i j k
be three vectors. A vector v in the plane of a
and
b
, whose projection on c
is 1/ 3 is given by
(a)
i 3j 3k (b) -
3i 3j k (c)
3i j 3k (d)
i 3j 3k
53. (c)
v i j k i j k -
v.c 1
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1
c 3
- - -
- 1
v i 2 1 j k 2 1
Only option (c) satisfies this one. 23
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
SECTION-II (TOTAL MARKS: 16)
(Multiple Correct Choice Type)
This section contains 4 multiple choice questions. Each question has four choices (a), (b), (c) and (d) outof
which ONE OR MORE may be correct.
54. Let f: R R be a function such that f ( x + y ) = f x f y , x, y R 7 8
If f(x) is differentiable at x = 0, then
(a) f(x) is differentiable only in a finite interval containing zero.
(b) f(x) is continuous 7 8x R
(c) f ' x is constant 7 8x R
(d) f(x) is differentiable except at finitely many points
54. (b) (c)
h 0 h 0 h 0
f x h f x f x f h f x f h f 0
f ' x Lim Lim Lim
h h h
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Put x y 0, f 0 0
f ' x f ' 0
f x f ' 0 x c
f x kx c, c = 0 as f 0 0
Function is continuous. f(x) is constant.
55. Let the eccentricity of the hyperbola
2 2
2 2
x y
1
a b
be reciprocal to that of the ellipse
2 2
x 4y 4 .
If the hyperbola passes through a focus of the ellipse, then
(a) the equation of the hyperbola is
2 2
x y
1
3 2
(b) a focus of the hyperbola is (2, 0)
(c) the eccentricity of the hyperbola is
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5
3
(d) the equation of the hyperbola is
2 2
x 3y 3
55. (b) (d)
2 2
x y
1
4 1
,
2 2 2 2 2
a 4, b 1,e 1 b / a e 3 / 2
eccentricity of hyperbola = 2 / 3
Focus of ellipse = 9 9 ae,0 3,0
2 2
2 2
x y
1
a b
passes through focus 9 3,0
2
a 3
2 2 2
b a e 1 1
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Equation of hyperbola:
2 2 2 2
2 2
x y x y
1 1
a b 3 1
2 2
x 3y 3
Focus of hyperbola : 9 9 ae,0 2,024
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
56. Let M and N be two 3 3 non singular skew symmetric matrices such that MN = NM. If P
T
denotes the transpose of P, then
1
2 2 T 1
M N M N MN T
is equal to
(a)
2
M (b)
2
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N (c)
2
M (d) MN
56. (c) Since
T T
M M, N N
1 T
2 2 T 1
M N M N MN
1 T 1 1
2 2 1 2 2 1 T
M N MN MN M N N M N M
1 1 1
M N M N M M MN MN M M 2 2
57. The vector (s) which is/are coplanar with vectors
i j 2k and
i 2 j k , and perpendicular to
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the vector
i j k is /are
(a)
j k (b) -
i j (c)
i j (d)
j k
57. (a) (d) Let
a i j 2k i 2 j k -
- - - i 2 j 2 k
a i j k 0
- a= j k
Taking
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1, a = -j + k
1, a= j - k
25
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
SECTION-III (TOTAL MARKS: 15)
(Paragraph Type)
This section contains 2 paragraphs. Based upon one of the paragraphs 2 multiple choice questions and
based on other paragraph 3 multiple choice questions have to be answered. Each of these has fourchoices
(a), (b), (c) and (d) out of which ONLY ONE is correct.
Paragraph for question Nos 58 and 59
Let U1
and U2
be two urns such that U1
contains 3 white and 2 red balls, and U2
contains only 1 white ball.
A fair coin is tossed. If head appears then 1 ball is drawn at random from U1
and put into U2
. However, if
tail appears then 2 balls are drawn at random from U1 and put into U2.Now 1 ball is drawn at randomfrom
U2.
58. The probability of the drawn ball from U2 being white is
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(a) 13/30 (b) 23/30 (c) 19/ 30 (d) 11/30
58. (b)
3 2 3 2
1 1 2 2
5 5 5
2 2 2
For appearing head For appearing tail
1 3 2 1 1 2 1 C C C C
P 1 + 1 23 / 30
2 5 5 2 2 3 3 C C C
) * ) *
59. Given that the drawn ball from U2 is white, the probability that head appeared on the coin is
(a) 17/23 (b) 11/23 (c) 15/23 (d) 12/23
59. (d)
Using Bayes theorem
P =
3 2 3 2
1 1 2 2
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5 5 5
2 2 2
1 3 2 1
1
2 5 5 2
12 / 23
1 3 2 1 1 2 1 C C C C
1 1
2 5 5 2 2 3 3 C C C
) *
) * ) *
Paragraph for question Nos 60 and 62
Let a, b and c be three real numbers satisfying
1 9 7
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a b c 8 2 7 0 0 0
7 3 7
) *
(E)
60. If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a +
b + c is
(a) 0 (b) 12 (c) 7 (d) 6
60. (d) D =
1 8 7
9 2 3 0
1 1 1
system has non trivial solution.
If c = k, b = -6/7 k, a = -k/7
2a + b+ c = 1
2k / 7 6k / 7 k 1 k 726
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
: P a,b,c 1,6, 7
7a b c 7 6 7 6
61. Let be a solution of
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62. Let b = 6, with a and c satisfying (E). If
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and 5 are the roots of the quadratic equation
2
ax bx c 0 , then
n
n 0
1 1
;
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5
< is
(a) 6 (b) 7 (c) 6/7 (d) ;
62. (b) b = 6, k = -7, c = - 7, a = 1
Q.E. becomes
2
x 6x 7 0; a + b = -6 , ab = -7
n n 2
n 0 n 0
6 6 6 1
1 ...... 7
7 7 7 6
1
7
; ;
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5
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2
f x x cx with c = 1.
2
f x x x . f 2 6
64. If z is any complex number satisfying z 3 2i 2 3 , then the minimum value of 2z 6 5i is
64. (5) Expression
5
2 Z 3 i
2
which is equivalent to double the distance between complex
number z and 3, 5 / 2 which lies on one the diameter x = 3 of circle given. Hence
minimum value of expression =
5
2 5
2
65. Let
1 2 100
a ,a ,.......a be an arithmetic progression with
1
a 3 and
p
p i
i 1
S a , 1 p 100
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5 6 5nd d
6 nd d
m
n
S
S
) *
independent of n when d = 6
2 1 a a d 928
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
66. Consider the parabola
2
y 8x . Let 1
be the area of the triangle formed by the end points of its
latus rectum and the point P
1
,2
2
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on the parabola, and 2
be the area of the triangle formed by
drawing tangents at P and at the end points of the latus rectum. Then
1
2
is
66. (2) Extremities of L.R. are (2,4) & (2, -4)
1 6
Points of intersection of tangents at these points are (-2, 0), (1, 3) & (-1, -1)
2 3
67. Let f
1 sin
sin tan
cos 2
, where
4 4
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# #
! ! . Then the value of
d
f
d tan
is .
67. (1) Put k =
1 sin sin
tan tan k
cos 2 cos 2
2
2 2
2 2
sin 1
tan k sec k 1
1 2sin cos ec 2
2
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2 2 2
2 2
cos ec 1 1
sec k sin k tan
cos ec 2 cot
sin k tan f sin k tan Hence.
d f
1
d tan
68. The minimum value of the sum of real numbers
5 4 3 8
a ,a ,3a ,1,a
and
10
a with a > 0 is
68. (8) Using A.M 6 G.M
5 4 3 3 3 8 10
a a a a a 1 a a
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(d) III in haematite and II, III in magnetite
3. (d) Hamatite is Fe2O3 where iron is in III oxidation state.
Magentite is Fe3O4 which is a mixed oxide where iron exhibits both II and III oxidation
state.
4. Among the following complexes (K-P)
K Fe CN K , CO NH Cl L , Na Co oxalate M , Ni H O ]Cl N ,K Pt
3
6 6 3 3 3 3 6 4 2 2 2 CN O
and
2 3 6 2
Zn H O NO P
the diamagnetic complexes are
(a) K,L,M,N (b) K,M,O,P (c) L,M,O,P (d) L,M,N,O
4. (c) K has one unpaired e
N all octahedral complexes are outer orbital complex 2
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5. Passing H2S gas into a mixture of
2 2 2
Mn , Ni ,Cu
and
2
Hg
ions in an acidified aqueous
solution precipitates
(a) CuS and HgS (b) MnS and CuS
(c) MnS and NiS (d) NiS and HgS
5. (a) CuS and HgS get precipitated
6. Consider the following cell reaction:
2 o
2Fe O 4H 2Fe 2H O E 1.67V s 2 g aq aq 2
At
2 3
Fe 10 M, P O 0.1 atm 2
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and pH = 3, the cell potential at
o
25 C is
(a) 1.47 V (b) 1.77 V (c) 1.87 V (d) 1.57 V
6. (d)
4
2 2
2
2
2
O H
.06 .06
1.67 log Fe log 1.55V
2 4 H O
7. The freezing point (in
o
C ) of a solution containing 0.1g of
2
6
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K Fe CN Mol.Wt.329
in 100g
of water
1
K 1.86K kg mol
f
is
(a)
2
2.3 10
(b)
2
5.7 10
(c)
3
5.7 10
(d)
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2
1.2 10
7. (a)
2
T 1.86 0.1 20 4 / 329 2.3 10
f
8. Amongst the compound given, the one that would form a brilliant colored dye on treatment with
NaNO2 in dil. HCl followed by addition to an alkaline solution of
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naphthol is
8. (c) 3
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SECTION II (TOTAL MARKS: 16)
(MULTIPLE CORRECT ANSWER(S) TYPE)
This section contains 4 multiple choice questions. Each question has four choices (a),(b),(c) and (d) out of
which ONE OR MORE may be correct.
9. For the first reaction
2 5 2 2
2N O g 4NO g O g
(a) The concentration of the reactant decreases exponentially with time.
(b) The half-life of the reaction decreases with increasing temperature.
(c) The half-life of the reaction depends on the initial concentration of the reactant.
(d) The reaction proceeds to 99.6% completion in eight half-half duration.
9. (abc)
1/2 1/2 t
693 2.3 100
log
t 8 t A
t
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100
2.4 log
A
t
A 100 / 250 0.4
99.6%
10. The correct functional group X and the reagent / reaction conditions Y in the following scheme are
(a) X COOCH ,Y H / Ni / heat 3 2
(b) X CONH ,Y H / Ni / heat 2 2
(c) X CONH ,Y Br / NaOH 2 2
(d) X CN,Y H / Ni / heat 2
10. (abcd) Alcohol gives polyster
NH2 group gives polyamide
11. The equilibrium
I o II
2Cu Cu Cu
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In aqueous medium at
o
25 C shifts towards the left in the presence of
(a) NO3
(b) Cl
(c) SCN
(d) CN
11. (bcd) Cl ,SCN and CN
form insoluble compound of CuCl,CuSCN and CuCN which
drags the equilibrium reaction in backward direction.
12. Reduction of the metal centre in aqueous permanganate ion involves
(a) 3 electrons in the neutral medium (b) 5 electrons in neutral medium
(c) 3 electrons in alkaline medium (d) 5 electrons in acidic medium
12. (ad)4
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SECTION-III (TOTAL MARKS: 24)
(INTEGER ANSWER TYPE)
This section contains 6 questions. The answer to each of the questions is a single-digit integer, ranging
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Cl 1.6 10 10 10
Cl 10
Ag 1.6 10 x 7
15. The number of hexagonal faces that are present in truncated octahedron is
15. (8)
16. The total number of contributing structures showing hyperconjugation (involving C-H bonds) for
the following carbocation is
16. (6) There are six hydrogens. 5
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B, darken the bubbles corresponding to q and r in the ORS.
19. Match the transformations in Column I with appropriate options in Column II
Column I Column II
(a) CO s CO g
2 2 (p) Phase transition
(b) CaCO s CaO s CO g
3 2 (q) Allotropic change
(c)
2
2H H g
(r)H is positive
(d) (s)
(t)
S is positive
S is negetive
19. a (p,r,s) b (r,s) c (t) d (q,r)
20. Match the reactions in Column I with appropriate types of steps/ reactive intermediate involved in
these reactions as given in Column II.
Column- I Column- II
(a) (p) Nucleopphilic substitution
(b) (q) Electrophilic substitution
(c) (r) Dehydration
(d) (s)
(t)
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Nucleophilic addition
Carbanion
20. a (r,s, t) b (s,p) c (s,r) d q,r 7
Quest Tutorials
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PHYSICS PAPER-II
PART II-PHYSICS
SECTION I (TOTAL MARKS: 24)
(SINGLE CORRECT CHOICE TYPE)
This Section contains 8 multiple choice questions. Each question ha four choice (a),(b),(c) and (d) out of
which ONLY ONE is correct.
21. The density of a solid ball is to be determined in an experiment. The diameter of the ball is
measured with a screw gauge, whose pitch is 0.5mm and there are 50divisions on the circular
sacle. The reading on the main scale is 2.5mm and that on the circular scale is 20 divisions. If the
measured mass of the ball has relative error of 2%, the relative percentage error in the density is
(a) 0.9% (b) 2.4% (c) 3.1% (d) 4.2%
21. (c) LC = (0.5/50)=10
2
mm
Reading = 2.5 + 20
10
2
= 2.7 mm
3
m
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/ m / m 3. d / d
4 d
3 2
2
2 3 10 / 2.7 100 3.1%
22. A wooden block performs SHM on a frictionless surface with frequency, 0
. The block carries a
charge +Q on its surface. If now a uniform electric filed E
is switched-on as shown, the SHM of
the block will be:
(a) of the same frequency and with shifted mean position
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(b) of the same frequency and with the same mean position
(c) of changed frequency and with shifted mean position
(d) of changed frequency and with the same mean position.
22. (a) Restoring force remains same, so frequency does not change.
Only mean position shifted rightward by (EQ/K) 8
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23. A light ray travelling in glass medium is incident on glass-air interface at an angle of incidence .
The reflected (R) and transmitted (T) intensities, both as function of , are plotted. The correct
sketch is
23. (c) for
C
0 T R
&
C
- T 0,R 100%
Also 0 T 100%,R 0%
24. A satellite is moving with a constant speed V in a circular orbit about the earth. An object of
mass m is ejected from the satellite such that it just escapes from the gravitational pull of the
earth. At the time of its ejection, the kinetic energy of the object is
(a)
2
(1 / 2)mV (b)
2
mV (c)
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2
3 / 2 mV (d)
2
2mV
24. (b)
2 2 2
mV / r GmM / r r GM / V e e
Now using energy conservation for particle
2
e
1/ 2 Mu GM m / r 0
2 e
e
2
1 GM
mu m 0
2 GM
V
2 2 1 / 2 Mu mV K.E9
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Quest Tutorials
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25. A long insulated copped wire is closely wound as a spiral of N turns. The spiral has inner radius
a and out radius b. The spiral lies in the X -Y plane and a steady current l flows through the
wire. The Z-component of the magnetic filed at the centre of the spiral is:
(a)
0Nl b
ln
2 b a a
(b)
0Nl b a
ln
2 b a b a
(c)
0Nl b
ln
2b a
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0Nl / 2 b a ln b / a
26. A point mass is subjected to two simultaneous sinusoidal displacement in x-direction,
1
x t A sin t ! and
2
x t A sin t 2 / 3 !
. Adding a third sinusoidal displacement
3
x t Bsin t ! " brings the mass to a complete rest. The values of B and " are
(a) 2 A, 3 / 4 (b) A, 4 / 3
(c) 3 A, 5 / 6 (d) A, / 3
26. (b)
1 2
x x A sin t A sin t 2 / 3 ! !
A 1 / 2 sin t 3 / 2 cos t ! !
! A sin t / 3
Now
1 2 3
x x x 0
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! ! " Asin t / 3 Bsin t
" B A & 4 / 310
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27. Which of the filed patterns given below is valid for electric field as well as for magnetic field?
27. (c) Electric field produced by the time varying magnetic field is circular.
For a long straight wire, the magnetic field lines are circles with their centres on the wire.
28. A ball of mass 0.2kg rests on a vertical post of height 5m. A bullet of mass 0.01kg, travelling with
a velocity
1
Vms
in a horizontal direction, hits the centre of the ball. After the collision, the ball
and bullet travel independently. The ball hits the ground at a distance of 20 m and the bullet ay a
distance of 100m from the foot of the post. The initial velocity V of the bullet is :
(a)
1
250ms
(b)
1
250 2 ms
(c)
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1
400ms
(d)
1
500ms
11
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28. (d) Time of flight after collision for both are
1
bullet
ball
t 2h / g 1sec
V 100 /1 100ms
V 20 /1 20ms 1
From conservation of Linear momentum
m u m v m v
bullet bullet bullet bullet ball ball
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, then the Gauss law will
still be valid
(b) The Gauss law can be used to calculate the field distribution around an electric dipole
(c) If the electric field between two point charges is zero somewhere, then the sign of the two
charges is the same.
(d) The work done by the external force in moving a unit positive charge from point A at potential
VA to point B at potential VB is V V B A
29. (cd) For point charge, flux for a sphere if point charge is placed at the centre
2
2.5
0
1
4 r
4 r
#
[According to question]
0 $ # q /
So Gauss law is not valid.
Electric field due to a dipole is not radially symmetric.
So Gauss law application does not give field distribution.
For electric field to be zero, directions of electric field must be opposite and equal in
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magnitude. This occurs only if sign of both the charges is same.
ext.agent elc.field B A
A B B A
W U U
W V V
q 1 q 1
30. A thin ring of mass 2kg and radius 0.5m is rolling without slipping on a horizontal plane with
velocity
1
1ms
. A small ball of mass 0.1kg moving with velocity
1
20ms
in the opposite direction,
hits the ring at a height of 0.75m and goes vertically up with velocity
1
10ms
. Immediately after
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the collision.
(a) the ring has pure rotation about its stationary CM
(b) the ring comes to a complete stop
(c) friction between the ring and the ground is to the left
(d) there is no friction between the ring and the ground 13
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
30. (ac) Applying CLM horizontal direction:
CoM ring
0.1 20 2 1 2V
V 0
CoM
Using CAM about contact point:
From given data, initial angular momentum is anticlockwise.
Lring finally
is anticlockwise
Hence friction on the ring acts leftward to oppose rotation.
31. A series R-C circuit is connected to AC voltage source. Consider two cases; (a) when C is without
a dielectric medium and b when C is filled with dielectric of constant 4. The current
R
I of the
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following is/are true ?
(a)
A B
R R
I I (b)
A B
R R
I I (c)
A B
V V C C
(d)
A B
V V C C
31. (bc) Z for 1
st
case,
2
2
A
1
Z R
C
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!
Z for 2
nd
case,
2
B 2
1
Z R
k c
!
So
B A
Z Z So, I I
B A R R
Since, V is same
source
V increases for B
Resistance
V decreases for B
Capacitor
32. Two solid spheres A and B of equal volumes but of different densities
A B
d and d are connected by
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a string. They are fully immersed in a fluid of density dF. They get arranged into an equilibrium
state as shown in the figure with a tension in the string. The arrangement is possible only if
(a)
A F
d d (b)
B F
d d (c)
A F
d d (d)
A B F
d d 2d
32. (abd) Considering the F.B.D
T = Vg d d
F A and T Vg d d B F
For both conditions, T > 0
F A d d and
B F
d d
Equating tension,
F B A
2d d d 14
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SECTION-III (TOTAL MARKS 24)
(INTEGER ANSWER TYPE)
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train f f
o
S ut 10cos 60 t S S 1.15
ball f f ball train
2
3u 5 3 3u 3 / 2 a 1.15 a 5 ms
34. Two batteries of different emfs and different internal resistances are connected as shown. The
voltage across AB in volts is
34. (5) If V is p.d. across A & B , i1 is current through 1% & i2 is current through 2% , then
V 6 i 3 2i
1 2
..(i)
1 2 i V 6 & i V 3 / 2
Also
1 2
i i 0 ..(ii)
V = 5V 15
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
35. Water (with refractive index = (4/3)) in a tank is 18 cm deep. Oil of refractive index (7/4) lies on
water making a convex surface of radius of curvature R = 6cm as shown. Consider oil to act as
thin lens. An object S is place 24 cm above water surface. The location of its image is at x cm
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max incident
& Maximum potential acquired (when maximum no. of photoelectrons emitted)
V K ne / r
max
S / unit & KE eV
max
max
7
n 15 /14.4 10
37. A series R C combination is connected to an AC voltage of angular frequency
1
500 radian s
! . If the impedance of the R C circuit is R 1.25 , the time constant (in
millisecond) of the circuit is
37. (4)
2
2 2 2 2 2
Z R X R 1.25 R 1 / C C
!
2 2
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R / 4 1 / 500C
2 2
RC 4 / 500
RC 1 / 250 s 4 ms16
Quest Tutorials
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38. A block of mass 0.18 kg is attached to a spring of force constant 2 Nm
-1
. The coefficient of
friction between the block and the floor is 0.1. Initially the block is at rest and the spring is un
stretched. An impulse is given to the block as shown in the figure. The block slides a distance of
0.06 m and comes to rest for the first time. The initial velocity of the block in ms
-1
is V = N/10.
Then N is
38. (4)
2 2
1 / 2 mV 1 / 2 Kx fx
2
2 2 2 2 1 2
1/ 2 18 10 V 1 / 2 2 6 10 18 10 10 10 ) 6 10
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2 V 144 / 900 N /10 12 / 30 N 417
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SECTION IV (TOTAL MARKS: 16)
(MATRIX MATCH TYPE)
This section contains 2 questions. Each question four statements (A, B, C and D) given in Column I and
five statements (p, q, r, s and t) in Column II. Any given statement in Column I can have correct matching
with ONE or MORE statement(s) given in Column II. For example, if for a given question, statement B
matches with the statements given in q and r, then for the particular question, against statement B,darken
the bubbles corresponding to q and r in the ORS.
39. One mole of a monatomic ideal gas is taken through a cycle ABCDA as shown in the P V
diagram. Column II gives the characteristics involved in the cycle. Match them with each of the
processes given in Column I.
Column I Column II
(a) Process A B (p) Internal energy decreases
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Q W U;
Q ve 0 ve
r, t is correct 18
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40. Column I shows four system, each of the same length L, for producing standing waves. The
lowest possible natural frequency of a system is called its fundamental frequency, whose
wavelength is denoted as
1
) . Match each system with statements given in Column II describing
the nature and wavelength of the standing waves.
Column I Column II
(A) Pipe closed at one end (p) Longitudinal waves
(B) Pipe open at both ends (q) Transverse waves
(C) Stretched wire clamped at
both ends
(r) )
f L
(D) Stretched wire clamped at
both ends and at mid point
(s)
f ) 2L
(t)
f ) 4L
40. A (p,t), B (p,s), C (q,s), D (q,r)
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A Longitudinal standing wave also,
f f
) ) / 4 L 4L p, t is correct
B Longitudinal standing wave also,
f f
) ) / 2 L 2L p,s is correct
C Longitudinal standing wave also,
f f ) ) / 2 L 2L q,s is correct
D Longitudinal standing wave also,
f f f
) ) ) / 2 / 2 L L q, r19
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
MATHEMATICS PAPER II
PART III MATHEMATICS
SECTION I (TOTAL MARKS: 24)
(SINGLE CORRECT ANSWER TYPE)
This section contains 8 multiple choice questions. Each question has four choices (a), (b), (c) and (d) outof
which ONLY ONE is correct.
41. Let f(x) = x
2
and g(x) = sin x for all x *R. Then the set of all x satisfying ( f o g o g o g ) (x) = (g
o g o f) (x), where (f o g) (x) = f (g (x)) , is
(a) + n , n * 0,1,2,....... (b) + n , n* 1,2,........
(c) 2 2n , n * ......, 2, 1,0,1,2........ (d) 2n , n ...., 2, 1,0,1,2...... *
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41. (a) (f o g o g o f ) (x) = ( g o g o f) (x)
2
2 2
sin sin x sin sin x
2 sin sin x 0 or 1
+ * x n , n 0,1,2......
42. Let (x, y) be any point on the parabola
2
y 4x . Let P be the point that divides the line segment
from (0, 0) to (x, y) in the ratio 1 : 3. Then the locus of P is
(a)
2
x y (b)
2
y 2x (c)
2
y x (d)
2
x 2y
42. (c) y = 2 x
Let P be (h, k) x = 4h, x 2k
2 k h
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44. A value of b for which the equations
2
x bx 1 0
2
x x b 0
have one root in common is
(a) - 2 (b) -i 3 (c) i 5 (d) 2
45. (b) be the common root
2
b 1 0
2
b 0
2
2 b 1
1 b
,
1 b
b 1
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3 2
b 3b 0 b 3b i 3
45. Let ! $1 be a cube root of unity and S be the set of all non singular matrices of the form
2
1 a b
1 c
1
, -
!
! !
, where each of a, b and c is either ! or
2
! . Then the number of distinct matrices in
the set S is
(a) 2 (b) 6 (c) 4 (d) 8
45. (a)
2
1 a b
1 c 0
1
! $
! !
2 ! ! $ 1 a c ac 0a c 1 $ ac 1 $
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Clearly a = ! or
2
! or c =
2
! ! or
but
2
c $ ! as it makes the determinant zero.
also
2
a $ !
2
! ! a , b or ! ! ,c
No of such matrices = 2
46. The circle passing through the point (-1, 0) and touching the y axis at (0, 2) also passes through
the point
(a) (-3/2, 0) (b) (-5/2,2) (c) (-3/2, 5/2) (d) (-4, 0)
46. (d) Let C be (h, 2)
Equation of circle:
circle passes through (-1, 0) , h = -5/2
Equation of circle will be (x + (5/2))
2
+ (y 2)
2
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= 25/4
Only (-4, 0) satisfies
47. If
1 x
2 2
x 0
Lim 1 x ln 1 b 2bsin
, b > 0 and * , then the value of is
(a) + 4 (b) + 3 (c) + / 6 (d) + / 2
47. (d)
2 2 2 2
1 b 2bsin b 2bsin 1 0
2 4
2sin 4sin 4
b
2
+
=
2 2 2
sin cos 1 sin +
For b to be real, cos 0 + 221
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48. Let f: . 1,2 0, be a continuous function such that f(x) = f( 1 x) for all x * 1,2 . Let
2
1
1
R xf x dx
, and R2 be the area of the region bounded by y = f(x), x = -1, x = 2, and the x
axis. Then
(a) R 2R 1 2 (b) R 3R 1 2 (c)
1 2
2R R (d)
1 2
3R R
48. (c)
2
1 2 1 1 2
1
R 1 x f 1 x dx R R 2R R
22
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Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
SECTION II (TOTAL MARKS: 16)
MULTIPLE CORRECT CHOICE TYPE
This section contains 4 multiple choice questions. Each question has four choices (a), (b), (c) and (d) outof
which ONE or MORE may be correct.
49. Let L be a normal to the parabola
2
y 4x . If L passes through the point (9, 6), the L is given by
(a) y x + 3 = 0 (b) y + 3x 33 = 0
(c) y + x 15 (d) y 2x + 12 = 0
49. (a,b,d) Equation of normal y = mx 2am am
3
i.e y = mx 2m m
3
Passing through (9, 6)
3
m 7m 6 0 m 3,2,1
Equation of normals are y + 3x 33 = 0, y 2x 12 0 , y x 3 0
50. Let f: 0,1 R be defined by f(x) =
b x
1 bx
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f ' x 0 x 0,1 as b 0,1
1 bx
/ * *
Also, f(x) is continuous for all x 0,1 * as domain of definition is R 1 / b where 1/b
> 1. Hence, f(x) is strictly decreasing, so invertible.
b x y b b x 1
f x y x f x f x
1 bx yb 1 1 bx
2
1
f ' b
b 1
and
2
f ' 0 b 1
Since , f ' x exist for all x * 0,1
1
f
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1
2
2
4
3
2
5
Clearly, f(x) is not differentiable at x = 0, f(x) is differentiable at x = 1 and f(x) is
differentiable x = -3/2 24
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
SECTION III (TOTAL MARKS: 24)
(INTEGER ANSWER TYPE)
This section contains 6 questions. The answer to each of the questions is a single digit integer, ranging
from 0 to 9. The bubble corresponding to the correct answer is to be darkened in the ORS.
53. Let
i
3
e
! , and a, b, c, x, y, z be non zero complex numbers such that
2
2
a b c X
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a b c y
a b c z
! !
! !
Then the value of
2 2 2
2 2 2
x y z
a b c
is
53. (3)
2 2 2
c c c
x y z xx yy zz
= (a + b+ c)
c c c 2 c c 2 c 2 c c c 2
a b c a b c a b c a b c a b c ! ! ! ! ! ! ! !
=
2 2 2
2 2 2
2 2 2
x y z
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3 a b c 3
a b c
54. Let y ' x y x g ' x g x g ' x , y 0 0, x R * , where f(x) denotes
df x
dx
and g(x) is a
given non constant differentiable function on R with g(0) = g(2) = 0. Then the value of y(2) is
54. (0) y ' x y x g ' x g x g ' x
Linear differential equation with integrating factor
g x
e
g x g x g x g x
' ' ' ' y x e g x g ' x e dx y x e e g x 1 c
Since, y 0 0 and g 0 0 c = 1
g x g 2
y x g x 1 e y 2 g 2 1 e 0
55. Let M be a 3 3
matrix satisfying
0 1
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M 1 2 ,
0 3
, - , -
1 1
M 1 1
0 1
, - , -
and
1 1
M 1 1
1 12
, - , -
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M 1 1
0 1
1 2 4 5 7 8
x x 1, x x 1, x x 1
1 4 7 x 0, x 3, x 2
1 2 3 4 5 6 7 8 9
1 0
M 1 M 0 x x x 0; x x x 0, x x x 12
1 12
3 6 9 x 1, x 5, x 7
0 1 1
M 3 2 5
2 3 7
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Sum of diagonal entries = 9
56. The straight line 2x 3y = 1 divides the circular region
2 2
x y 6 2 into two parts. If
3 5 3 1 1 1 1
S 2, , , , , , , *
4 2 4 4 4 8 4
1 6
4 7
5 8
Then the number of point (s) in S lying inside the smaller part is
56. (2) 2x 3y = 1 meets co ordinate axes at
1 1 5 3
,0 & 0, ,
2 3 2 4
'
lies outside the circle
hence ruled out.
For,
3
x 2, y
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4
: 2x 3y 1 0
For,
1 1
x , y
4 4
: 2x 3y 1 0
For,
1 1
x , y
8 4
: 2x 3y 1 < 0
3 1 1
2, & ,
4 4 4
are the only two points
57. Let
a i k, b = -i + j
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and
c i 2 j 3k
be three given vectors. If r
is a vector such that
r b c b
and r a 0 '
, then the value of r b 0 '
is 26
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
57. (9) r c b 0
) ) ) r c b 1 j 2 j 3k
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If
r xi yj zk
, then r.a 0 x z 0 x z
Also, x 1 , y = 2 + , z = 3 -3 = 1- = 4 ) ) ) )
r 3j 6 j 3k
' r b 3 6 9
58. The number of distinct real roots of
4 3 2
x 4x 12x x 1 0 is
58. (2) Let us assume that all four roots are real and distinct
Hence, f(x) = 0 must have 3 distinct real roots and
f " x 0 must have 2 distinct real roots, but that is not true as
2
f x 12 x 2x 2 99 with D < 0
Hence, f(x) = 0 cant have all four roots real
As, f(0) = -1, f(1) =9 and f(-1) = 15
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f (x) 0 must have two distinct roots one in (-1, 0) and the other one in (0, 1) 27
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
SECTION II (TOTAL MARKS: 16)
(MULTIPLE CORRECT ANSWERS TYPE)
This section contains 2 questions. Each question has four statements (A, B, C and D) given in Column I
and five statements (p, q , r, s and t ) in Column II. Any given statement in Column I can have correct
matching with ONE and MORE statement(s) given in Column II. For example, if for a given question,
Statement B matches with the statements given in q and r, then for the particular question, againststatement
B, darken the bubbles corresponding to q and r in the ORS.
59. Match the statements given in Column I with the values given in Column II
Column I Column II
(A)
If
a j 3k,b j 3k
and
c 2 3k
from a triangle, then the
internal angle of the triangle
between a
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and b
be
(p)
6
(B)
If
b
2 2
a
f x 3x dx a b
, then
the value of f
6
is
(q) 2
3
(C)
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The value of
5
2 6
7
6
sec x dx
ln 3
is
(r)
3
(D) The maximum value of
1
Arg
1 z
for z 1, z 1 $ is
given by
(r)
(t)
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2
59. A q , B p , C s , D t
For (A): Since, side lengths are 2, 2, 2 3 , hence angle between a
and b
is
4 4 12
cos 1/ 2 2 / 3
2 2 2
' '
For (B) :
b
2 2
a
1
f x dx b a f x x f / 6 / 6
2
For (C) :
5/6
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7 /6
1
ln sec x tan x ln 3 ln
ln 3 ln 3 3
, -
For (D) :
i
2 2
1 1 cos isin
z e ,
1 z 1 cos sin
1 sin
arg f
1 z 1 cos
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, which is maximum when 228
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
60. Match the statements given in Column I with the intervals/ union of intervals given in Column II
Column I Column II
(A) The set
2
2iz
Re : z is complex number, z 1, z 1
1 z
1 6
4 $ + 7
5 8
(p) . : . , 1 1,
(B) The domain of the function
x 2
1
2 x 1
8 3
f x sin is
1 3
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(q) . : . ,0 0,
(C)
If
1 tan 1
f tan 1 tan
1 tan 1
, then the set
f : 0
2
1 6
4 7 2
5 8
is
(r) 2,.
(D)
If
3/2
f x x 3x 10 , x 0 - , then f(x) is
increasing in
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(s) . : . , 1 1,
(t) . : . ,0 2,
60. A s Let k =
i
2 2i 2
2iz 2ie 2i cos isin
Re Re Re
1 z 1 e 2sin 2isin cos
=
2i cos isin 1
Re
2isin cos isin sin
It is defined only when k * . : . , 1 1,
B t Since,
x 2
2x 2
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8 3
1 1
1 3
'
2 2
x
2 x
8.3
1 1
9 3
2 2
Put
x
y 3
2
8y
1 1
9 y
2 2
2
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8y y 9 y 1
1 0
9 y y 3 y 3
2 -
y 9
0
y 3
-
( y + 1 and y + 3 are always (+) ve)
y 3 and y 9 -* . : . x ,1 [2, )
and
2
8y y 9 y 1
1 0
9 y y 3 y 3
2 -
y 1
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0
y 3
-
2 y 1 and y 3 29
Quest Tutorials
H-108, New Asiatic Building, Connaught place, New Delhi 110001
* . : . x ( ,0] 2,
The common solution is x * . : . ( ,0] 2, .
C r
Since,
2
1 tan 1
f tan 1 tan 2sec
1 tan 1
* . f [2, )
D r Let f(x) =
3/2
x 3x 10
1/2 3/2
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f ' x 3 / 2 x 3x 10 x 3
For increasing,
f ' x 0 -
- 3 x 3x / 2 5 x 0
' - - 3 x 5x 10 0 x 2